Happens all the time in the workforce so why would it not work in sports?
I don't believe that's correct. It's relatively common for a change in CEO to herald a change in leadership team composition/membership. It's also relatively common for unsuccessful applicants for corporate leadership roles to depart having been unsuccessful.
Perhaps you're equating the Head Coach role to some sort of middle-management position, where a bunch of applicants might apply including some who report to that position, and they may stay despite being unsuccessful. But I don't think this situation is analogous to that.
This is the key leadership role in the organisation. The top dog. And my view is as above. A vanquished top dog won't be welcome, and won't want, to hang around to be the top dog's seventh assistant.
I've seen it a few times once a CFO became the temporary CEO and applied for the job, they didn't get it but stayed on board. I've also seen a number of senior managers apply for a CEO position and ultimately another applicant was chosen, they all stayed.
It also happens frequently, as you say, in middle management.
I don't know many people who would be petty enough to let the fact that a better candidate got a job that they wanted reflect on their performance if they stayed, if they felt that way they would leave.
You want the best employees possible and if they were in the position before are happy to stay in a lesser role then who cares?